CSE 780 Algorithms Advanced Algorithms Graph Alg. DFS Topological sort

CSE 780 Algorithms Objectives On completion of this lecture, students should be able to: 1. Write a dfs algorithm 2. Apply dfs in topological sort and finding strongly- connected components.

CSE 780 Algorithms Breadth-first Search Works for both directed and undirected graph Starting from source node s, visits remaining nodes of graph from small distance to large distance Produce a BF-tree Return distance between s to any reachable node in time O(|V| + |E|)

CSE 780 Algorithms Depth-first Search Breadth-first search: Go as broad as possible at each node Depth-first search (a different strategy): Go as deep as possible first

CSE 780 Algorithms More Formally Again, a node white: unvisited gray: discovered but not finished black: finished (explored) For every node v  V d[v]: time v is discovered f[v]: time v is finished (all edges in v’s adjacency list are explored) f[v] - d[v] = time from grey to black

CSE 780 Algorithms Pseudo-code Time complexity

CSE 780 Algorithms Example

CSE 780 Algorithms Depth-first Forest Consider all edges (p(u), u) When it happens: p(u) is grey, u is white A forest, called Depth-first forest Depends on the order of vertices Example from previous page Property: Start (finish) time for each tree: same order as if we visit nodes in pre-order (post-order) tree walk

CSE 780 Algorithms Properties of DFS Parenthesis Theorem: Any two nodes u and v, one of the following 3 cases: (1) u is descendant of v, and d[v] < d[u] < f[u] < f[v] (2) v is descendant of u, and d[u] < d[v] < f[v] < f[u] (3) u is not descendant of v, neither is v a descendant of u, and d[u] < f[u] < d[v] < f[v] or d[v] < f[v] < d[u] < f[u] u is descendant of v iff d[v] < d[u] < f[u] < f[v] White-path Theorem u is descendant of v iff at the time of d[v], there is a all white path from v to u.

CSE 780 Algorithms Classification of Edges Four types of an edge (u,v) Tree edge Non-tree edges: Back edge: u is descendant of v Forward edge: v is descendent of u Cross edge: others Distinguish by the color of v Example b dc e f g h j a k

CSE 780 Algorithms Theorem In a DFS of an undirected graph G, every edge of G is either a tree edge or a back edge. b dc e f g h j a k

CSE 780 Algorithms Connected Components Undirected graph DF forests represents the set of connected components Directed graph Does the tree rooted at u include all nodes accessible from u ?

CSE 780 Algorithms Directed Acyclic Graph DAG: directed acyclic graph Determine whether a directed graph is DAG or not How? A directed graph G is a dag iff a DFS of G yields no back edges. How about an undirected acyclic graph?

CSE 780 Algorithms Topological Sort A topological sort of a dag G = (V, E) A linear ordering A of all vertices from V If edge (u,v)  E => A[u] < A[v] undershort s pants belt shirt tie jacke t shoes socks watch

CSE 780 Algorithms Topological Sort Using DFS Topological-Sort(G) Call DFS(G) to compute finishing times f[v] for each v Output vertices in descreasing order of finishing time Time complexity O (|V| + |E|)

CSE 780 Algorithms Example undershort s pants belt shirt tie jacke t shoes socks watch 11, 16 12, 15 6, 7 3, 4 2, 5 1, 8 13, 14 17, 18 9, 10 socks, undershots, pants, shoes, watch, shirt, belt, tie, jacket

CSE 780 Algorithms Correctness Theorem Topological-Sort(G) produces a topological sort of a directed acyclic graph G Proof: Consider any edge (u, v) Goal: f[v] < f[u] Three possible types of edges: Tree edge, forward edge, cross edge

CSE 780 Algorithms Connected Components An undirected graph is connected if every pair of vertices is connected by a path A graph that is not connected is naturally decomposed into several connected components. A connected components is a maximal set of vertices, which are all connected.

CSE 780 Algorithms Connected Components (cont.) The graph below has 3 connected components {1,2,5}, {3,6} and {4}. An undirected graph is connected if it has exactly one connected component.

CSE 780 Algorithms Strongly-connected components

CSE 780 Algorithms Strongly-connected components The above graph has 3 strongly connected components {1,2,4,5}, {3} and {6} A directed graph is strongly connected if it has only one strongly connected component.

CSE 780 Algorithms Transpose of a Graph The transpose of a directed graph G = (V,E) is the graph G T = (V, E T ) where E T = {(u,v): (v,u) Є E}. E T consists of the edges of G with their directions reversed. G and G T have exactly the same strongly connected components.

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CSE 780 Algorithms

Corollary Let C and C’ be distinct strongly connected components in directed graph G = (V,E). Suppose that there is an edge (u,v)  E T, where u  C and v  C’. Then f(C) < f(C’) Note: f(C) = max u  C {f[u]}

CSE 780 Algorithms Theorem STRONGLY-CONNECTED-COMPONENTS(G) correctly computes the strongly connected components of a directed graph G. Proof. We argue by induction that the vertices of each tree form a strongly connected component. The inductive hypothesis: The first k trees produced in line 3 are strongly connected.

CSE 780 Algorithms The basis for the induction, when k=0, is trivial. The inductive step: assume that each of the first trees produced is a strongly connected component. Consider the (k+1)st tree. Let the root of this tree be vertex u, and let u be in a strongly connected component C. Because of how we choose roots in line 3, f[u] = f(C) > f(C’) for any strongly connected component C’ that has yet to be visited.

CSE 780 Algorithms By inductive hypothesis, at the time that the search visits u, all other vertices of C are white. Therefore, all other vertices of C are descendants of u in its tree. Moreover, by inductive hypothesis and corollary 22.15, any edge in G T that leave C must be to strongly connected components that have already visited. Thus, no vertex in any strongly connected component other than C will be a descendant of u during the depth-first search of G T. Thus, the vertices of the depth-first tree in G T that is rooted at u form exactly one strongly connected component.

CSE 780 Algorithms Summary Graph search/traversal method Breadth-first search Discover nodes in shortest distance (# links) e.g, chess Depth-first search Parenthesis theorem e.g, topological sort